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This work addresses several aspects of the dependence on p of the higher eigenvalues λ n to the Robin problem, - Δ p u = λ | u | p - 2 u x ∈ Ω , | ∇ u | p - 2 ∂ u ∂ ν + b | u | p - 2 u = 0 x ∈ ∂ Ω. Here, Ω ⊂ R N is a C 1 bounded domain, ν is the outer unit normal, Δ p u = div (| ∇ u | p - 2 ∇ u) stands for the p-Laplacian operator and b ∈ L ∞ (∂ Ω) . Main results concern: (a) the existence of the limits of λ n as p → 1 , (b) the 'limit problems' satisfied by the 'limit eigenpairs', (c) the continuous dependence of λ n on p when 1 < p < ∞ and (d) the limit profile of the eigenfunctions as p → 1 . The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios. [ABSTRACT FROM AUTHOR]

Image forgery detection is essential for forensic and security applications. In local locations where key points (KPs) are few, traditional approaches that rely on them suffer from restrictions. To tackle this problem, this paper suggests an innovative strategy that combines pre-processing, feature extraction, and classification methods. We specifically present EFP-AEHO, or Enhanced Flower Pollination-Adaptive Elephant Herd Optimization, for modifying the Deep Belief Network's (DBN) weights. Our method provides better performance by first converting RGB images into the Lab color space, then applying Eigenvalue Asymmetry (EAS) to extract texture features. Its efficacy is highlighted by experimental findings on the F2000 dataset, which show testing accuracy of 99.4%, TPR of 98%, and TNR of 99.12%. [ABSTRACT FROM AUTHOR]

Image forgery detection is essential for forensic and security applications. In local locations where key points (KPs) are few, traditional approaches that rely on them suffer from restrictions. To tackle this problem, this paper suggests an innovative strategy that combines pre-processing, feature extraction, and classification methods. We specifically present EFP-AEHO, or Enhanced Flower Pollination-Adaptive Elephant Herd Optimization, for modifying the Deep Belief Network's (DBN) weights. Our method provides better performance by first converting RGB images into the Lab color space, then applying Eigenvalue Asymmetry (EAS) to extract texture features. Its efficacy is highlighted by experimental findings on the F2000 dataset, which show testing accuracy of 99.4%, TPR of 98%, and TNR of 99.12%. [ABSTRACT FROM AUTHOR]

We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the eigenvalue of Courant sharp eigenfunctions that is independent of the widths of the necks. [ABSTRACT FROM AUTHOR]

In this paper, we consider the wreath product of two graphs that yields much more larger graphs than other graph products. We drive a formula for the adjacency spectrum of the wreath product of a complete graph and a circulant graph. As its applications, we obtain their (normalized) Laplacian spectra and further compute the adjacency spectra when the circulant graphs are respectively the Möbius ladder graph, the crown graph and the Andrásfai graph. Moreover, some known results are generalized and a problem is posed for further study. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied. [ABSTRACT FROM AUTHOR]

For a connected graph G, let A(G) be the adjacency matrix of G and D(G) be the diagonal matrix of the degrees of the vertices in G. The A α -matrix of G is defined as A α (G) = α D (G) + (1 - α) A (G) for any α ∈ [ 0 , 1 ]. The largest eigenvalue of A α (G) is called the A α -spectral radius of G. In this article, we characterize the graphs with maximum A α -spectral radius among the class of unicyclic and bicyclic graphs of order n with fixed girth g. Also, we identify the unique graphs with maximum A α -spectral radius among the class of unicyclic and bicyclic graphs of order n with k pendant vertices. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new estimate for the (product) γ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly γ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674). [ABSTRACT FROM AUTHOR]

In this work, we propose a piecewise Jacobi–Gauss spectral collocation (JGSC) method for simulating a multi-particle system involving processing delay. Through the use of Jacobi orthogonal approximation and simple Picard iteration, the method obtains the Jacobi series solution of the multi-particle model, allowing us to derive the numerical solution of the processing delay directly. The matrix–vector form of the method helps to obtain the solution parallelly and improves the efficiency significantly. Additionally, the eigenvalues of the iteration's coefficient matrix are evaluated in order to analyze the convergence of the JGSC method numerically. Numerical experiments illustrate that the JGSC method can keep the high accuracy and efficiency. Furthermore, simulation results of the model indicate that the flocking behavior can only achieved with small enough processing time delays and large enough sizes of the neighborhood. [ABSTRACT FROM AUTHOR]

We consider N×N$N\times N$ non‐Hermitian random matrices of the form X+A$X+A$, where A$A$ is a general deterministic matrix and NX$\sqrt {N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by N1+o(1)$N^{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1)$N^{1+o(1)}$; both results are optimal up to the factor No(1)$N^{o(1)}$. The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the N$N$‐dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near‐optimal lower tail estimate for the small singular values of X+A−z$X+A-z$, is of independent interest. [ABSTRACT FROM AUTHOR]

The energy of a graph is calculated by summing the absolute values of the eigenvalues found in its adjacency matrix. In this study, we present examples of line graphs with energy equivalent to the energy of a complete graph, which are called the borderenergetic graphs. As the examples of borderenergetic line graphs are rare, we introduce a new type of graphs called almost borderenergetic graphs whose energy differs from the borderenergetic energy by at most 1. In this paper, we begin by exploring the energy properties of line graphs derived from regular graphs and strongly regular graphs. Specifically, we establish a criterion for a line graph to exhibit borderenergetic characteristics when it consists of p many connected regular graphs and q many complete graphs, even when the original regular graph is disconnected. Additionally, we present a criterion for the line graph of a strongly regular graph to exhibit borderenergetic characteristics. To illustrate these concepts, we offer examples of connected borderenergetic graphs that are not necessarily complete. In the second part, we give the spectrum of the line graph of rK1∇K2, and show that it is an almost borderenergetic graph for r ≥ 5. [ABSTRACT FROM AUTHOR]

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems. [ABSTRACT FROM AUTHOR]

The topological physics has sparked intensive investigations into topological lattices in photonic, acoustic, and mechanical systems, powering counterintuitive effects otherwise inaccessible with usual settings. Following the success of these endeavors in classical wave dynamics, there has been a growing interest in establishing their topological counterparts in diffusion. Here, we propose an additional real-space dimension in diffusion, and the system eigenvalues are transformed from "imaginary" to "real." By judiciously tailoring the effective Hamiltonian with coupling networks, localized and delocalized topological modes are realized in heat transfer. Simulations and experiments in active thermal lattices validate the effectiveness of the proposed theoretical strategy. This approach can be applied to establish various topological lattices in diffusion systems, offering insights into engineering topologically protected edge states in dynamic diffusive scenarios. [ABSTRACT FROM AUTHOR]

In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems. [ABSTRACT FROM AUTHOR]

This paper aims to employ the Darboux transformation (DT) to discover the interaction solutions of the Zakharov equation (Eq. (1.2) for δ=1). Through partial degradation of eigenvalues, interaction solutions of the model are constructed on the basis of high-order breather solutions. The study derives interaction solutions involving breather and b-positon solutions through partial degradation of eigenvalues (λj→λ1). Further, interaction solutions comprising breather and lump solutions are obtained through partial double degradation of eigenvalues (λj→λ0). Then, several interaction solutions containing b-positon and lump solutions are extracted through mixed degradation of eigenvalues (λj→λ1 and λk→λ0). In particular, the dynamic evolution characteristics of these solutions are studied. It is believed that these studies make a significant contribution to the understanding of the Zakharov equation and its possible applications in physics. [ABSTRACT FROM AUTHOR]

We prove new multiplicity results for some critical growth p$p$‐biharmonic problems in bounded domains. More specifically, we show that each of the problems considered here has arbitrarily many solutions for all sufficiently large values of a certain parameter λ>0$\lambda > 0$. In particular, the number of solutions goes to infinity as λ→∞$\lambda \rightarrow \infty$. We also give an explicit lower bound on λ$\lambda$ in order to have a given number of solutions. This lower bound will be in terms of an unbounded sequence of eigenvalues of a related eigenvalue problem. Our multiplicity results are new even in the semilinear case p=2$p = 2$. The proofs are based on an abstract critical point theorem. [ABSTRACT FROM AUTHOR]

Summary: We investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B‐splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B‐spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary‐fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B‐splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B‐splines, even in the boundary‐fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B‐spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one‐dimensional elastic‐plastic bar impacting a rigid wall is illustrative. [ABSTRACT FROM AUTHOR]

Flat, nondispersive bands and topological phase transition in multiple Su–Schrieffer–Heeger (SSH) chains, cross-linked via periodically arranged nodal points are explored within a tight binding framework. We give an analytic prescription, based on a real space decimation scheme, which extracts the energy eigenvalues corresponding to the flat bands along with their degeneracy. The topological phase transition is confirmed through the existence of quantized Zak phase for all the Bloch bands, and the edge states that are protected by chiral symmetry, consistent with the bulk-boundary correspondence. In addition to the edge states, the entangled systems are shown to give rise to clusters of localized eigenstates in the bulk of the system, in contrast to a purely one-dimensional SSH system. [ABSTRACT FROM AUTHOR]

We consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable. [ABSTRACT FROM AUTHOR]

Let Ω = Ω0∖Θ¯ ⊂ ℝn, n ≥ 2, where Ω0 and Θ are two open, bounded and convex sets such that Θ¯ ⊂ Ω0 and let β < 0 be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to Ω, with Robin boundary condition on ∂Ω0 and Neumann boundary condition on ∂Θ. In [47] it is proved that the spherical shell is the only maximizer for the first Robin–Neumann eigenvalue in the class of domains Ω with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed. [ABSTRACT FROM AUTHOR]

The spectral Dirichlet problem is considered in a waveguide made of a semi-infinite cylinder Π and the resonator ΘR obtained by inflating R times a fixed star-shaped domain Θ. The behavior of the scattering coefficient s(R) is studied as the parameter R grows, namely, it is verified that this coefficient moves clockwise without stops along the unit circle in the complex plane. For s(R) = -1, the proper threshold resonance occurs; it is accompanied by the appearance of an almost standing wave and provokes various near-threshold anomalies, in particular, splitting eigenvalues off from the threshold. Under the geometrical symmetry, resonances of other type are shown to be generated by trapped waves at the threshold. The justification of asymptotics is made by applying the technique of weighted spaces with detached asymptotics and an analysis of the singularities of physical fields at the edge ∂ΘR ∩ ∂Π. [ABSTRACT FROM AUTHOR]

Given a |$g$| -dimensional abelian variety |$A$| over a finite field |$\mathbf{F}_{q}$|⁠ , the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most |$g$|⁠. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for |$g \le 3$|⁠. We also give a partial classification for simple ordinary abelian varieties of prime dimension |$g\geq 3$|⁠. [ABSTRACT FROM AUTHOR]

Non-cooperative targets, such as birds and unmanned aerial vehicles (UAVs), are typical low-altitude, slow, and small (LSS) targets with low observability. Radar observations in such scenarios are often complicated by strong motion clutter originating from sources like airplanes and cars. Hence, distinguishing between birds and UAVs in environments with strong motion clutter is crucial for improving target monitoring performance and ensuring flight safety. To address the impact of strong motion clutter on discriminating between UAVs and birds, we propose a frequency correlation dual-SVD (singular value decomposition) reconstruction method. This method exploits the strong power and spectral correlation characteristics of motion clutter, contrasted with the weak scattering characteristics of bird and UAV targets, to effectively suppress clutter. Unlike traditional clutter suppression methods based on SVD, our method avoids residual clutter or target loss while preserving the micro-motion characteristics of the targets. Based on the distinct micro-motion characteristics of birds and UAVs, we extract two key features: the sum of normalized large eigenvalues of the target's micro-motion component and the energy entropy of the time–frequency spectrum of the radar echoes. Subsequently, the kernel fuzzy c-means algorithm is applied to classify bird and UAV targets. The effectiveness of our proposed method is validated through results using both simulation and experimental data. [ABSTRACT FROM AUTHOR]

Domain generalization (DG) aims to learn knowledge from multiple related domains to achieve a robust generalization performance in unseen target domains, which is an effective approach to mitigate domain shift in remote sensing image classification. Although the sharpness-aware minimization (SAM) method enhances DG capability and improves remote sensing image classification performance by promoting the convergence of the loss minimum to a flatter loss surface, the perturbation loss (maximum loss within the neighborhood of a local minimum) of SAM fails to accurately measure the true sharpness of the loss landscape. Furthermore, its variants often overlook gradient conflicts, thereby limiting further improvement in DG performance. In this paper, we introduce implicit sharpness-aware minimization (ISAM), a novel method that addresses the deficiencies of SAM and mitigates gradient conflicts. Specifically, we demonstrate that the discrepancy in training loss during gradient ascent or descent serves as an equivalent measure of the dominant eigenvalue of the Hessian matrix. This discrepancy provides a reliable measure for sharpness. ISAM effectively reduces sharpness and mitigates potential conflicts between gradients by implicitly minimizing the discrepancy between training losses while ensuring a sufficiently low minimum through minimizing perturbation loss. Extensive experiments and analyses demonstrate that ISAM significantly enhances the model's generalization ability on remote sensing and DG datasets, outperforming existing state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper investigates the coupled Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in two-dimensional bounded domains with a focus on global well-posedness and the formation of striking patterns, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time in the case of subcritical mass. Concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method. While studying the global existence and the concentration phenomenon of the chemotaxis–fluid model, we develop the global W 2 , p theory for the 2D stationary Stokes system subject to Navier boundary conditions with zero friction and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem. [ABSTRACT FROM AUTHOR]

Isospectral surfaces provide a rich family of benchmark problems. In this paper the efficacy of a hp -finite element method in such Laplace-Beltrami eigenvalue problems has been studied. In addition, for the p -version natural auxiliary space error estimators have been shown to be effective also in this context. As part of the numerical experiments further numerical evidence of the validity of the so-called Quarter Sphere Conjecture has been produced. For the isospectral surface pairs constructed via transplantation, the extension for perforated surfaces is derived. The lack of such constraints for surfaces with isothermal coordinates is also demonstrated. [ABSTRACT FROM AUTHOR]

In this study, the energy control and asymptotic stability of the 1D sine-Gordon equation were investigated from the viewpoint of numerical approximation. An order reduction method was employed to transform the closed-loop system into an equivalent system, and an average-central finite difference scheme was constructed. This scheme is not only energy-preserving but also possesses uniform stability. The discrete multiplier method was utilized to obtain the uniformly asymptotic stability of the discrete systems. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established. Finally, several numerical experiments based on the eigenvalue distribution of the linearized approximation systems were conducted to demonstrate the effectiveness of the numerical approximating algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors. [ABSTRACT FROM AUTHOR]

Aiming to improve the robustness and adaptiveness of the recently investigated capped norm linear discriminant analysis (CLDA), this paper proposes a regularized linear discriminant analysis based on the generalized capped l 2 , q -norm (GCLDA). Compared to CLDA, there are two improvements in GCLDA. Firstly, GCLDA uses the capped l 2 , q -norm rather than the capped l 2 , 1 -norm to measure the within-class and between-class distances for arbitrary q > 0 . By selecting an appropriate q, GCLDA is adaptive to different data, and also removes extreme outliers and suppresses the effect of noise more effectively. Secondly, by taking into account a regularization term, GCLDA not only improves its generalization ability but also avoids singularity. GCLDA is solved through a series of generalized eigenvalue problems. Experiments on an artificial dataset, some real world datasets and a high-dimensional dataset demonstrate the effectiveness of GCLDA. [ABSTRACT FROM AUTHOR]

Let A be a nonnegative irreducible matrix of order n and A i (t) be the matrix obtained by increasing its i th diagonal entry by a positive number t. An index i ∈ [ n ] = { 1 , ... , n } is called a universal winner for A , if, letting ρ (⋅) denote the spectral radius, ρ (A i (t)) ≥ ρ (A j (t)) for j ∈ [ n ] and for t ∈ (0 , ∞). Let G be a strongly connected digraph with vertex set [ n ]. Let W G be the set of all nonnegative matrices whose underlying graph structure is G. We say a vertex u structurally dominates another vertex v in G , if ρ (A u (t)) ≥ ρ (A v (t)) for all t ∈ (0 , ∞) and for all A ∈ W G. We characterize the class of digraphs G that do not have a vertex that structurally dominates all other vertices. We say two vertices u and v structurally tie if they structurally dominate each other in G. We supply an equivalent graph theoretic condition for the structural tying of two vertices in G. Let S ⊆ [ n ] be nonempty. We characterize the class of strongly connected digraphs G with vertex set [ n ] such that S is the set of universal winners for each A ∈ W G. We also characterize the strongly connected digraphs whose vertex set can be partitioned into subsets P 1 , P 2 , ... , P k such that vertices inside a part P i structurally tie with each other and vertices of P i structurally dominate vertices of P j strictly (without tying) for i < j. [ABSTRACT FROM AUTHOR]

An eigenvalue λ of a signed graph S of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. Characterizing signed graphs with exactly k (1 ≤ k ≤ n) main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Z. Stanić has noticed that a signed graph has exactly one main eigenvalue if and only if it is net-regular, and in this paper, we study signed graphs with exactly two distinct main eigenvalues by studying (0 , 1 , 2) -multi-graphs with exactly two distinct main eigenvalues. We partially characterize the case when the basic graphs of the (0 , 1 , 2) -multi-graphs are trees, and propose some problems for further research. [ABSTRACT FROM AUTHOR]

A graph is claw-free if it does not contain a star of order 4 as an induced subgraph. In this paper, we characterize all claw-free connected graphs with all but four eigenvalues equal to 0 or − 1. [ABSTRACT FROM AUTHOR]

In this paper, the Riemann‐Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first‐ and second‐order dark soliton solutions on the nonzero background under the reflectionless condition. [ABSTRACT FROM AUTHOR]

A nonlinear parabolic equation with transformation of the spatial variable and periodic conditions on the circle is considered. Using the method of separation of variables, we obtain properties of eigenfunctions and eigenvalues of the corresponding linearized problem. Using the method of central manifolds, we prove the existence and stability of spatially inhomogeneous stationary solutions. Based on the Galerkin method, we analyze approximate solutions of the original problem. [ABSTRACT FROM AUTHOR]

This study revisits the freely vibrating problem of asymmetric L-shaped membranes using a three-segmented domain decomposition (3-SDD) strategy. Motivated by the need for more accurate and flexible methods, the 3-SDD strategy is compared with the previously proposed two-segmented domain decomposition (2-SDD) strategy. The region-matching technique is used to derive an alternative series solution, and the eigenvalues obtained are compared with those in existing research. The convergence behavior and sensitivity analyses reveal that the 3-SDD strategy offers improved accuracy and stability, particularly for higher truncation terms. Detailed comparisons of the first four eigenvalue squares show strong agreement between the 3-SDD and 2-SDD strategies, confirming the reliability of both methods. This research establishes a foundation for the vibration analysis of complex membrane structures, emphasizing the benefits of the 3-SDD approach for upcoming engineering applications and showcasing its potential for broader applicability in practical scenarios. The findings underscore the importance of utilizing multi-segmented decomposition strategies to enhance the accuracy and flexibility of free vibration analysis. [ABSTRACT FROM AUTHOR]

To address the challenges in establishing the state transfer matrix and the complexity of eigenvalue calculation in determining the multi-parameter stability boundaries of high-order nonlinear Vienna rectifiers, a novel numerical computation method is proposed in this paper. This method leverages a numerical stability criterion and a grid variable step search to efficiently calculate these stability boundaries. The small-signal model of the Vienna rectifier is derived by constructing the time-varying state transfer matrix using the periodic solution of the harmonic balance method. Eigenvalues are rapidly calculated via the periodic numerical solution of the state transfer matrix. The proposed parameter sensitivity-based grid variable step search method ensures a fast and accurate determination of stability boundaries. A hardware experimental setup is established to validate the stability boundaries of the Vienna rectifier under various parameter variations, including load, component, and control changes. The experimental results closely match the simulations, confirming the correctness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

We present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others. [ABSTRACT FROM AUTHOR]

Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution. [ABSTRACT FROM AUTHOR]

A predator-prey model with prey-taxis under homogeneous Dirichlet boundary conditions was established. Based on the prior estimate of positive solutions, the asymptotic stability of trivial and semi-trivial solutions was obtained by using the eigenvalue theory of linearization operators and spectral theory. The existence of positive steady-state solutions was established using the fixed point index theory in a cone. By analyzing two related eigenvalues, the coexistence region with respect to the growth rates of two species was obtained. The results show that prey-taxis has an obvious influence on the steady-state solutions. [ABSTRACT FROM AUTHOR]

The hybrid active power filter (HAPF) can effectively suppress the 75–125 Hz low-frequency harmonic current generated by the EAST poloidal field power supply. However, the low-frequency harmonic current suppression performance of HAPF is greatly affected by its control parameters when the circuit parameters are determined. In this paper, the state–space equation of the HAPF system is established, and the relationship between the eigenvalues of the state–space matrix and the HAPF control parameters is analyzed using the Lyapunov method. Based on the variation law of the eigenvalues, by optimizing the control parameters of the HAPF system, the low-frequency harmonic current suppression performance of HAPF is improved by 41.9%. [ABSTRACT FROM AUTHOR]

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain Ω ⊂ R N are also proved. As application, we measure explicitly how small has to be diam (Ω) so that weak and strong maximum principles associated to this problem hold in Ω . [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian. [ABSTRACT FROM AUTHOR]

We provide a variant of Geršgorin's circle theorem, where the ℓ1$\ell ^1$‐estimates are swapped for ℓ2$\ell ^2$‐estimates, more suitable for the infinite‐dimensional Hilbert space setting. [ABSTRACT FROM AUTHOR]

This work examines the global dynamics of classical solutions of a two‐stage (juvenile–adult) reaction–diffusion population model in time‐periodic and spatially heterogeneous environments. It is shown that the sign of the principal eigenvalue λ∗$\lambda _*$ of the time‐periodic linearized system at the trivial solution completely determines the persistence of the species. Moreover, when λ∗>0$\lambda _*>0$, there is at least one time‐periodic positive entire solution. A fairly general sufficient condition ensuring the uniqueness and global stability of the positive time‐periodic solution is obtained. In particular, classical solutions eventually stabilize at the unique time‐periodic positive solutions if either each subgroup's intrastage growth and interstage competition rates are proportional, or the environment is temporally homogeneous and both subgroups diffuse slowly. In the latter scenario, the asymptotic profile of steady states with respect to small diffusion rates is established. [ABSTRACT FROM AUTHOR]

A signed graph Σ = (G , σ) consists of an underlying graph G = (V , E) with a sign function σ : E → { − 1 , 1 } . Let A (Σ) be the adjacency matrix of Σ and λ 1 (Σ) denote the largest eigenvalue (index) of Σ. Define (K n , H −) as a signed complete graph whose negative edges induce a subgraph H. In this paper, we focus on the following question: which spanning tree T with a given number of pendant vertices makes the λ 1 (A (Σ)) of the unbalanced (K n , T −) as large as possible? To answer the question, we characterize the extremal signed graph with maximum λ 1 (A (Σ)) among graphs of type (K n , T −) . [ABSTRACT FROM AUTHOR]

This study introduces a 4 × 4 matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable equations, illustrating the integrable hierarchy. [ABSTRACT FROM AUTHOR]

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we "enhance" the definition of the virtual element space to make the L 2 -projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance. [ABSTRACT FROM AUTHOR]

Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin–Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus the supremum norm. [ABSTRACT FROM AUTHOR]

The paper discusses systems with viscoelastic elements that exhibit repeated eigenvalues in the eigenvalue problem. The mechanical behavior of viscoelastic elements can be described using classical rheological models as well as models that involve fractional derivatives. Formulas have been derived to calculate first- and second-order sensitivities of repeated eigenvalues and their corresponding eigenvectors. A specific case was also examined, where the first derivatives of eigenvalues are repeated. Calculating derivatives of eigenvectors associated with repeated eigenvalues is complex because they are not unique. To compute their derivatives, it is necessary to identify appropriate adjacent eigenvectors to ensure stable control of eigenvector changes. The derivatives of eigenvectors are obtained by dividing them into particular and homogeneous solutions. Additionally, in the paper, a special factor in the coefficient matrix has been introduced to reduce its condition number. The provided examples validate the correctness of the derived formulas and offer a more detailed analysis of structural behavior for structures with viscoelastic elements when altering a single design parameter or simultaneously changing multiple parameters. [ABSTRACT FROM AUTHOR]